Reflection positivity for the circle group

30th International Colloquium on Group Theoretical Methods in Physics

Neeb, K.-H.^1 ; Olafsson, G.^2

Department Mathematik, FAU Erlangen-Nurnberg, Cauerstrasse 11, Erlangen

91058, Germany^1

Department of Mathematics, Louisiana State University, Baton Rouge

LA

70803, United States^2

关键词: Analytic continuation;    Euclidean;    Modular data;    Parameter groups;    Positive definite function;    Reflection positivity;    Sesquilinear form;    Unitary representations;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/597/1/012004/pdf
DOI  :  10.1088/1742-6596/597/1/012004

来源: IOP

【 摘 要 】

In this note we characterize those unitary one-parameter groups (Utc)t∈R which admit euclidean realizations in the sense that they are obtained by the analytic continuation process corresponding to reflection positivity from a unitary representation U of the circle group. These are precisely the ones for which there exists an anti-unitary involution J commuting with Uc. This provides an interesting link with the modular data arising in Tomita-Takesaki theory. Introducing the concept of a positive definite function with values in the space of sesquilinear forms, we further establish a link between KMS states and reflection positivity on the circle.